: At the time of the screenshot, she's not just a hologram, she's intimately
: connected with the control room and it makes sense that she would attain
: some sort of power via that connection.

I think that it's just a holographic representation of the index. The Master Chief inserted the Index into it's little slot on the control panel, so Cortana would have to have removed it physically. What I always thought she did with it was absorb it into herself.